---
tag: 52N6S8WXM9
kind: definition
name: Brownian bridge
msc: [60J65, 60G15]
defines: 3MCVMBAGD1
uses: [22WEM6RTW0, 0KBFAWYRTE, 552AWX3QXN]
sources:
  - url: https://en.wikipedia.org/wiki/Brownian_bridge
---

## Definition

A *Brownian bridge* is a real stochastic process $(X_t)_{0 \le t \le 1}$ such that:

1. $(X_t)$ is a [[Gaussian process|22WEM6RTW0]] with $E[X_t] = 0$ for all $t \in [0, 1]$.
2. Its [[covariance|0KBFAWYRTE]] is $E[X_s X_t] = \min(s, t) - st$ for all $s, t \in [0, 1]$.
3. Almost surely, the path $t \mapsto X_t$ is continuous.

## Remarks

If $W$ is a [[Wiener process|552AWX3QXN]], then $X_t = W_t - t W_1$ is a Brownian bridge.
Check: $X$ is a linear image of the Gaussian process $W$, hence Gaussian and centred, with continuous paths, and $E[X_s X_t] = \min(s, t) - st - st + st = \min(s, t) - st$.
The Wiener process $W$ restricted to $[0, 1]$ satisfies clauses 1 and 3 but fails clause 2, since $E[W_s W_t] = \min(s, t)$; for instance $\operatorname{Var}(W_1) = 1$, while clause 2 forces $X_1 = 0$.
Clause 2 gives $X_0 = X_1 = 0$ almost surely and $\operatorname{Var}(X_t) = t(1 - t)$.
The law of a Brownian bridge is the conditional law of $(W_t)_{0 \le t \le 1}$ given $W_1 = 0$, in a sense made precise by a limit.